Let T:V->W be a linear transformation of two finite dimensional vector spaces V,W (both over a field F).

vballa15ei

vballa15ei

Answered question

2022-09-08

Let T : V W be a linear transformation of two finite dimensional vector spaces V, W (both over a field F).
My assignment is to show that there exists a basis B in V and a basis C in W such that the transformation F with respect to the bases B and C actually does have A as a transformation matrix.
Since V, W are vector spaces of course they have bases and I can do a transformation with respect to the bases from V to W. But how do I show that (some specific?) bases exist such that the transformation with respect to these bases have a given a A as a transformation matrix?
Assume A is transformation matrix of T. Then we take one vector in V expressed in (some basis) B and when we transform it to some vector in W, expressed in (some basis) 𝐶. So A does at least change the basis of a vector, but I since V and W might be of different dimension I don't really know where to go from here. Am I one the wrong track?

Answer & Explanation

nirosoh9

nirosoh9

Beginner2022-09-09Added 16 answers

Assume dim𝑉=𝑛 and dim𝑊=𝑛, then, by Steinitz's Theorem, 𝑉 and 𝑊 have bases with 𝑛 and 𝑚 elements respectively. Elements which you can write as the columns of two matrices, say 𝐵 and 𝐶. Now, think about how 𝐵 and 𝐶 are and how do they interact with 𝐴.

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